Question

(a) Suppose you have two quarters and a dime in your left pocket and two dimes and three quarters in your right pocket. You select a pocket at random and from it a coin at random. What is the probability that it is a dime? (b) Let \(x\) be the amount of money you select. Find \(E(x)\) (c) Suppose you selected a dime in (a). What is the probability that it came from your right pocket? (d) Suppose you do not replace the dime, but select another coin which is also a dime. What is the probability that this second coin came from your right pocket?

Short answer

(a) \(\frac{11}{30}\) (b) 0.3533 (c) \(\frac{6}{11}\) (d) \(60\to .\x}

Step-by-step solution

- Understanding the Total Coin Compositions

Identify the total number of coins in each pocket. Left pocket has 2 quarters and a dime, which totals 3 coins. Right pocket has 2 dimes and 3 quarters, which totals 5 coins.

- Probability of Selecting Each Pocket

Since each pocket is equally likely to be chosen, the probability of selecting either pocket is \(\frac{1}{2}\).

- Probability of Selecting a Dime from Left Pocket

The probability of selecting a dime from the left pocket is the number of dimes divided by the total number of coins in the left pocket, which is \(\frac{1}{3}\).

- Probability of Selecting a Dime from Right Pocket

The probability of selecting a dime from the right pocket is the number of dimes divided by the total number of coins in the right pocket, which is \(\frac{2}{5}\).

- Total Probability of Selecting a Dime

Using the law of total probability, the overall probability of selecting a dime is \(\frac{1}{2} \times \frac{1}{3} + \frac{1}{2} \times \frac{2}{5} = \frac{1}{6} + \frac{1}{5} = \frac{11}{30}\).

- Calculating Expected Value of x (E(x))

Let D be the event of picking a dime and Q be the event of picking a quarter. The expected value, \(E(x)\), is calculated as follows: \(E(x) = P(D) \times 0.10 + P(Q) \times 0.25\). Using the earlier result \(E(x) = \frac{11}{30} \times 0.10 + \frac{19}{30} \times 0.25 = \frac{11}{300} + \frac{95}{300} = \frac{106}{300} = 0.3533\).

- Probability that Dime is from Right Pocket Given Dime is Selected

Using Bayes' Theorem, the probability the selected coin is a dime from the right pocket is given by \(\frac{P(\text{Right Pocket}) \times P(\text{Dime from Right Pocket})}{P(\text{Dime})} = \frac{\frac{1}{2} \times \frac{2}{5}}{\frac{11}{30}} = \frac{\frac{2}{10}}{\frac{11}{30}} = \frac{6}{11}\).

- Second Coin Probability Calculation

Now, considering the case where a dime is not replaced and another dime is selected, the configuration of the right pocket is 1 dime and 3 quarters, and the left pocket remains unchanged. Using similar steps as above and calculating the required conditional probability, it comes to \( \frac{\frac{1}{2} \times \frac{1}{4}}{\frac{5}{30}} = \frac{1}{4}}{1/6} = \frac{3}{2 = 60%\).

Key concepts

Conditional Probability

Conditional probability is an important concept in probability theory.
It's the probability of an event occurring given that another event has already occurred. To calculate conditional probability, we use the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Here, P(A|B) is the probability of event A occurring given that B has happened.
P(A \cap B) is the probability of both A and B occurring.
And, P(B) is the probability that event B occurs.
In our exercise, we use conditional probability to find out the probability of selecting a dime from the right pocket given that a dime was selected.

Expected Value

Expected value, denoted as E(X), is the average value or mean of a random variable over a large number of experiments.
It's calculated using the formula:
\[ E(X) = \sum_{i} P(X_i) X_i \]
Where X_i represents the values the random variable can take, and P(X_i) is the probability of X_i occurring.
In our exercise, we calculate the expected value of the amount of money selected.
We consider the probabilities of selecting a dime (worth \(0.10) and a quarter (worth \)0.25).
The expected value provides us with an average monetary value expected from randomly picking a coin from either pocket.
This helps in decision-making and understanding the overall outcome we can expect in the long run.

Bayes' Theorem

Bayes' Theorem is a powerful tool in probability theory.
It helps us update the probability of a hypothesis based on new evidence.
The formula is:
\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]
Here, P(A|B) is the probability of event A given B.
P(B|A) is the probability of event B given A.
P(A) and P(B) are the probabilities of A and B, respectively.
In our exercise, we use Bayes' Theorem to determine the probability that a selected dime came from the right pocket.
It's essential for updating probabilities based on partial information.
This concept is widely used in various fields, including statistics, data science, and machine learning.

Law of Total Probability

The Law of Total Probability provides a way to calculate the total probability of an event based on its conditional probabilities.
The formula is:
\[ P(A) = \sum_{i} P(A|B_i) P(B_i) \]
Where P(A) is the total probability of event A occurring.
P(A|B_i) is the conditional probability of A given B_i.
P(B_i) is the probability of each of the mutually exclusive events B_i.
In our exercise, we applied the Law of Total Probability to find the overall probability of selecting a dime.
We consider the probabilities of selecting each pocket and then the probability of selecting a dime within those pockets.
This law ensures we consider all possible scenarios and their associated probabilities.